Scientific seminar on the differential and functional differential equation under the guidance of Professor A.L. Skubachevskii

Speaker: Malygina Vera V., candidate of physical and mathematical sciences, associate professor, leading researcher, Functional Differential Equations Research Center, Perm National Research Polytechnic University, Perm, Russia.

Topic: Stability of linear non-autonomous functional differential equations.

We investigate a class of linear differential equations with several variable delays and coefficients. The boundaries of coefficients and the maximum permissible values of delays are considered as parameters that define a certain family of equations.

We propose a method for studying the asymptotic properties of solutions to equations of the indicated family. The method is based on the construction of an auxiliary equation (so-called “test-equation”), whose stability guarantees the stability of all equations of the family.

The necessary and sufficient conditions for the exponential and uniform stability of the family are obtained in terms of properties of the solution to the test-equation.

For families of equations with a small number of independent parameters, we find the analytical form and the geometric interpretation of the boundaries of stability regions. For the simplest non-autonomous equation with one delay, the obtained stability criterion coincides with the famous “3/2-theorem” by A. D. Myshkis.